What Uses Does Lagrange's Interpolation See Today? More concretely, in what situations are the predictions made by Lagrange's Interpolation accurate? Say, if we have 4 known values of some country's populations, would the unknown value predicted by interpolating the points be accurate? I doubt it would, because population doesn't have to increase/decrease to fit an equation predicted by Lagrange. In what cases would the prediction be accurate?
 A: The basic Lagrange interpolation with many points is primarily of theoretical interest.
Polynomials are pretty well-behaved and well-understood functions, and it is sometimes useful to know while conducting a proof that there exists a well-behaved function that goes through any finite number of points you care to specify.
In practice, however, Lagrange interpolation on real-world data sets with many points fails horribly. Even tiny rounding or measurement errors in the middle of the sequence will blow up into insane oscillations of the interpolated function at the ends.
With a few points given -- no more than about 4, I'd say -- Lagrange interpolation can be useful for producing an approximation of the underlying function. It is probably misleading to think of it as a prediction, though. But if you already know that the underlying curve is nice and smooth throughout the interval you're looking at, then a polynomial interpolation can give you an arithmetic expression for something that follows that curve "pretty well". This gives you some advantages:


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*The arithmetic expression is often cheaper to evaluate than it is to find a value of the true curve.

*The arithmetic expression can be manipulated with symbolic algebra/calculus if the curve is input to some further computation, again saving effort.


The disadvantage is of course that the approximation may not follow the true curve exactly -- especially if it turns out that the true curve is not really as smooth and uniform as the approximation assumes.
It is a trade-off in each case whether the ease of computing with the approximation outweighs the cost of the approximation errors.
And, of course, if you do decide to approximate, a naive polynomial interpolation may not be the best way to do that. There are many other options -- some of which use polynomial interpolation as a component of a more intricate scheme.
A: If you know the value of some function at certain points only, and you want to know its value at some other point, then your only option is some sort of interpolation.
If you have some knowledge of the specific form of the function you expect, you can use that. You can fit a function of this specific form to your data, and then use that function for interpolation.
If you have no knowledge of the form of the function, then you may as well assume that it's something simple and easy to compute. Lagrange interpolation and spline interpolation satisfy this requirement. The only thing wrong with Lagrange interpolation is that the curves tend to get extraneous "wiggles" if you're not careful.
To answer your question, I'd say that Lagrange interpolation (or, more generally, polynomial interpolation) is quite widely used.
You can make the interpolation as accurate as you like just by increasing the degree of the polynomial. There's a system called "chebfun" that's very good at this.
